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Mirrors > Home > MPE Home > Th. List > tgbtwnxfr | Structured version Visualization version GIF version |
Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
Ref | Expression |
---|---|
tgcgrxfr.p | ⊢ 𝑃 = (Base‘𝐺) |
tgcgrxfr.m | ⊢ − = (dist‘𝐺) |
tgcgrxfr.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgcgrxfr.r | ⊢ ∼ = (cgrG‘𝐺) |
tgcgrxfr.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnxfr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnxfr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnxfr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnxfr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnxfr.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgbtwnxfr.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
tgbtwnxfr.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
tgbtwnxfr.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Ref | Expression |
---|---|
tgbtwnxfr | ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrxfr.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgcgrxfr.m | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tgcgrxfr.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgcgrxfr.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 762 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐺 ∈ TarskiG) |
6 | simplr 792 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ 𝑃) | |
7 | tgbtwnxfr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
8 | 7 | ad2antrr 762 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐸 ∈ 𝑃) |
9 | tgbtwnxfr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | 9 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐷 ∈ 𝑃) |
11 | tgbtwnxfr.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
12 | 11 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐹 ∈ 𝑃) |
13 | simprl 794 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ (𝐷𝐼𝐹)) | |
14 | eqidd 2623 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐷 − 𝐹) = (𝐷 − 𝐹)) | |
15 | eqidd 2623 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝑒 − 𝐹) = (𝑒 − 𝐹)) | |
16 | tgcgrxfr.r | . . . . . 6 ⊢ ∼ = (cgrG‘𝐺) | |
17 | tgbtwnxfr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
18 | 17 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐴 ∈ 𝑃) |
19 | tgbtwnxfr.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
20 | 19 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐵 ∈ 𝑃) |
21 | tgbtwnxfr.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
22 | 21 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐶 ∈ 𝑃) |
23 | simprr 796 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉) | |
24 | 1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23 | trgcgrcom 25423 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
25 | tgbtwnxfr.2 | . . . . . . . . 9 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | |
26 | 25 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
27 | 1, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26 | cgr3tr 25424 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
28 | 1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27 | trgcgrcom 25423 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐷𝑒𝐹”〉) |
29 | 1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28 | cgr3simp1 25415 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐷 − 𝐸) = (𝐷 − 𝑒)) |
30 | 1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28 | cgr3simp2 25416 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐸 − 𝐹) = (𝑒 − 𝐹)) |
31 | 1, 2, 3, 5, 8, 12, 6, 12, 30 | tgcgrcomlr 25375 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐹 − 𝐸) = (𝐹 − 𝑒)) |
32 | 1, 2, 3, 5, 10, 6, 12, 8, 10, 6, 12, 6, 13, 13, 14, 15, 29, 31 | tgifscgr 25403 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝑒 − 𝐸) = (𝑒 − 𝑒)) |
33 | 1, 2, 3, 5, 6, 8, 6, 32 | axtgcgrid 25362 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 = 𝐸) |
34 | 33, 13 | eqeltrrd 2702 | . 2 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐸 ∈ (𝐷𝐼𝐹)) |
35 | tgbtwnxfr.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
36 | 1, 2, 3, 16, 4, 17, 19, 21, 9, 7, 11, 25 | cgr3simp3 25417 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
37 | 1, 2, 3, 4, 21, 17, 11, 9, 36 | tgcgrcomlr 25375 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
38 | 1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37 | tgcgrxfr 25413 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) |
39 | 34, 38 | r19.29a 3078 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 〈“cs3 13587 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 cgrGccgrg 25405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 |
This theorem is referenced by: lnxfr 25461 tgfscgr 25463 legov 25480 legov2 25481 legtrd 25484 mirbtwni 25566 cgrabtwn 25717 cgrahl 25718 |
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